Relationship Between Hyperbolas and Hyperbolic Spaces

Question

I am trying to understand the difference between Hyperbolic Functions and Hyperbolic Spaces.

In my very limited knowledge of mathematics, I have only come across:

• Hyperbolas : https://i.ytimg.com/vi/Iu-4-fizlD4/maxresdefault.jpg

• Hyperbolic (Trigonometric) Functions: https://en.wikipedia.org/wiki/Hyperbolic_functions

Recently, I came across "Hyperbolic Spaces" (https://en.wikipedia.org/wiki/Hyperbolic_space). I tried to read more about this and I think I understand the general idea:

Although I don't fully understand the math, but I think a "hyperbola" would resemble the shape of a "hourglass cone" in 3 dimensions. As the picture above shows, a triangle in Euclidean Space becomes "distorted" when projected onto a Hyperbolic Space. This is my naïve guess about the relationship between hyperbolic functions and hyperbolic spaces.

Regarding all this, I have the following questions:

1) Although the sum of all 3 angles of the triangle is preserved when it is projected from Euclidean Space to Hyperbolic Space - is there a way to "mathematically understand the projection"? I know that the general equation of a Hyperbola is f(x) = 1/x : if we know the equations that make a specific triangle, how can we use our knowledge of the general equation of a Hyperbola to describe the projection?

2) When searching for images of "Hyperbolic Spaces", the following types of images always come up:

What is the relationship between the above diagrams and hyperbolic spaces? Are these pictures trying to illustrate some concept in particular (e.g. the projection of some shape from Euclidean Space to Hyperbolic Space, e.g. dodecahedral tessellation)? Is there any reason that these types of pictures are often used to illustrate the concept of Hyperbolic Spaces?

Thank you!

Answer 1

Before you read this, if you are confused about the notion of hyperbolic space, consider discarding prior ways you imagined hyperbolic spaces while reading this answer to minimize confusion.

The $d$-dimensional hyperbolic space $H^d$ is a simply connected smooth $d$-dimensional Riemannian manifold which has constant negative metric curvature everywhere. Two-dimensional hyperbolic space $H^2$ is called the hyperbolic plane. A sphere is in a way the opposite of hyperbolic space: it has constant positive curvature everywhere.

To begin with, I should make you aware of a common pitfall, by curvature I mean metric curvature, which is the curvature intrinsic to the material, unrelated to the way the material may or may not be embedded in a higher-dimensional space. For example, the metric curvature of a sheet of paper is constantly zero, no matter how you bend or fold it in space. When I talk of distances, I also only consider distances within the manifold, not within any possibly enclosing space.

Let's consider only the hyperbolic plane $H^2$ for now, with constant curvature -1, and try to compare it to the unit sphere $S^2$ and the flat plane $E^2$. For now, please don't try to imagine the entire hyperbolic plane visually as some surface within 3D space, it won't go well and it is in fact impossible without some form of creasing. A hyperboloid is decidedly not a hyperbolic plane.

In $E^2$, it is well-known that a circle with radius $r$ has circumference $2 \pi r$. On the sphere $S^2$, a circle with radius $r$ (always measured on the surface) has circumference $2\pi\sin(r)$. In the hyperbolic plane $H^2$, the circle with radius $r$ has circumference $2\pi\sinh(r)$.

Observe that $\sinh(r) = \frac 12(e^r + e^{-r})$ grows really quickly in terms of $r$, so in hyperbolic space circles quickly become very large. For example, in hyperbolic space the circle of radius $20$ has circumference $\approx 1500000000$, which illustrates nicely why imagination fails a bit here.

Now you probably know that you can't draw a flat map of Earth without distorting the size and shape of many features. and for similar reasons you can't draw a flat map of hyperbolic space without a lot of distortion. With such distortion, it is in fact possible to draw a map of all of hyperbolic space on a flat piece of paper, in fact several types of maps depending on the projection you use.

The first image you linked, with the red-and-white triangles, is a map of the hyperbolic plane using the Beltrami-Klein projection, which projects the entire hyperbolic plane into a euclidean disk. All the red-and-white triangles you see look different because of the projection, but they all have the same size and shape in their hyperbolic reality. A nice feature of the Beltrami-Klein projection is that everything drawn as a straight line is actually a hyperbolic straight line, so the edges of these triangles are actually straight.

The third image you linked, with the fish, uses a different projection called the Poincaré projection. Again, all fish are in reality the same size and shape. The white lines crisscrossing the image are actually straight, but the distortion caused by the Poincaré projection causes them to not be drawn as straight lines. However, the Poincaré projection preserves angles between lines, so all the $60^\circ$ angles are actually drawn as $60°$ angles. In the Klein projection angles would not be drawn to scale.

The second image you linked is not of the hyperbolic plane, but an image from within 3D hyperbolic space. You see the wire mesh of a tessellation of 3D hyperbolic space made from right-angled dodecahedra: regular dodecahedra where every angle is 90°. The perspective here is not some particular projection, but native perspective, just as if you were in the middle of hyperbolic space and took a photo.

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Let's talk about hyperbolic trigonometry and how it applies to hyperbolic space. We've already seen the first application earlier: circumferences of circles in terms of their radius. Integrating over the radius, we see that the area of a circle of radius $r$ is $\pi\cosh(r)$.

Another intrinsic application: imagine a cart travelling in a straight line on the hyperbolic plane, and a handle of length $x$ sticking out from the cart perpendicularly to the direction of travel. The end of this handle does not travel in a straight line, but on what is known as an equidistant curve or hypercycle, and the end of this handle travels at $\cosh(x)$ times the velocity of the cart.

Hyperbolic trig functions also have prominent roles in computations regarding hyperbolic triangles: given a triangle in $H^2$ with side lengths $a, b, c$ and angles $A, B, C$, you have the law of sines: $$\frac {\sin A}{\sinh a} = \frac {\sin B}{\sinh b} = \frac {\sin C}{\sinh c}$$ and the law of cosines with dual: $$\cosh c = \cosh a \cosh b - \sinh a \sinh b \cos C$$ $$-\cos C = \cos A \cos B - \sin A \sin B \cosh c$$ as well as a few other theorems seen here.

Hyperbolic trigonometric functions also have lots of use cases in computations within particular projections, for example in the Beltrami-Klein projection the distance between the origin and a point with map coordinates $(x,0)$ is equal to $\tanh^{-1}(x)$.

Finally, let's talk about hyperbolas and hyperboloids. The hyperboloid model of a hyperbolic space is another projection, this time drawn not on a flat surface but actually on one sheet of the two-sheeted hyperboloid. I'm not going to go into too much detail here, but this model is arguably the most useful model for practical applications because isometries of hyperbolic space correspond to linear maps in this projection, and hyperbolic trig functions, while not actually necessary to define or use the model, play a huge role in this model in terms of computing distances and isometries. Just be aware that this model also has distortion: after all, a two-sheeted hyperboloid has positive metric curvature and the hyperbolic plane has negative metric curvature.

Answer 2

I think the best way start is to first review some spherical geometry and stereographic projection, if you haven't already. The unit sphere is the surface $$ S = \{ x^2 + y^2 + z^2 = 1 \} $$ The shortest curve between any two points is called a geodesic segment. It is a piece of a great circle. Great circles are the intersections of planes through the origin with the sphere. The spherical distance between two points on the sphere is the angle between the two unit vectors from the origin to the points. Stereographic projection projects the sphere into the horizontal plane through the origin using rays from the north pole. There should be a picture of what geodesic triangles look like in the plane using stereographic projection, but I haven't found any.

The hyperboloid model of hyperbolic space is a negatively curved analogue of the sphere. It is the surface $$ H = \{ x^2 + y^2 - z^2 = -1 \}. $$ In the same way you define the angle between two unit vectors using the functions $\cos\theta$ and $\sin\theta$, you can define the angle of two vectors from the origin to two points on $H$ using the hyperbolic functions $\cosh \theta$ and $\sinh\theta$ and use this to define the hyperbolic distance between any two points on $H$. The geodesics (analogues of the great circles) are again the intersections of planes through the origin with $H$.

Stereographic projection projects $H$ onto not the plane but the unit disk in the $z=0$ plane using rays starting from $(0,0,-1)$. This results in the Poincaré disk model of hyperbolic space. This is depicted in the rightmost picture. The white stripes appear to be thickened geodesics. A geodesic in this model is any circular arc or straight line that intersects the boundary of the disk at a right angle.

The leftmost picture appears to be of the Klein model, which is obtained by projecting into the unit disk in the $z=1$ plane using rays from the origin. Here, it is obvious that geodesics are straight lines, because they are intersections of the disk with planes through the origin.

I'm not sure what the middle picture is showing. Since at least some of the curves appear to be circular arcs, it might be showing geodesic polygons in the Poincaré disk.